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Examples offrame bundle
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Theframebundleof a smooth manifold is the one associated to its tangent bundle.
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This example is from Wikipedia and may be reused under a CC BY-SA license.A motivating example of a solder form is the tautological or fundamental form on theframebundleof a manifold.
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This example is from Wikipedia and may be reused under a CC BY-SA license.In the modern approach, this is closely related to the definition of affine connections on theframebundle.
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This example is from Wikipedia and may be reused under a CC BY-SA license.This gives an equivalent more geometric way of describing the connection in terms of lifting paths in the manifold to paths in theframebundle.
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This example is from Wikipedia and may be reused under a CC BY-SA license.Along a curve in the surface, the connection defines a first order differential equation in theframebundle.
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This example is from Wikipedia and may be reused under a CC BY-SA license.This enabled the curvature properties of the surface to be encoded in differential forms on theframebundleand formulas involving their exterior derivatives.
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This example is from Wikipedia and may be reused under a CC BY-SA license.From any vector bundle, one can construct theframebundleof bases which is a principal bundle (see below).
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This example is from Wikipedia and may be reused under a CC BY-SA license.It can be constructed by a method entirely analogous to that of the ordinaryframebundle.
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This example is from Wikipedia and may be reused under a CC BY-SA license.The projection onto this subspace is defined by a differential 1-form on the orthonormalframebundle, the connection form.
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This example is from Wikipedia and may be reused under a CC BY-SA license.These expressions are differential 2-forms on the total space of aframebundle.
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This example is from Wikipedia and may be reused under a CC BY-SA license.All bundles associated with aframebundleare natural.
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This example is from Wikipedia and may be reused under a CC BY-SA license.At the same time, gauge gravitation theory exemplifies field theory on a principalframebundlewhose gauge symmetries are general covariant transformations which are not elements of a gauge group.
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This example is from Wikipedia and may be reused under a CC BY-SA license.More generally still, we can consider frame bundles associated to any vector bundle, or even arbitrary principal fiber bundles.
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This example is from Wikipedia and may be reused under a CC BY-SA license.Tangent and frame bundles over a world manifold are natural bundles characterized by general covariant transformations.
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This example is from Wikipedia and may be reused under a CC BY-SA license.These examples are from corpora and from sources on the web. Any opinions in the examples do not represent the opinion of the Cambridge Dictionary editors or of Cambridge University Press or its licensors.